# Math Help - Circular Curves and Tangents Lines

1. ## Circular Curves and Tangents Lines

Hi MHF
I'm having some hard times with circular functions like

Let $A$ be a point on the curve $C: X^2+Y^2-2X -4=0$. If the tangent line to $C$ at $A$ passes through point $P(4,3)$, then the length of AP is:

It seems hard to even see the image in my mind, I'm confused!

2. x² + y² - 2x - 4 = 0 is equivalent to (x - 1)² + y² = √5².
Line passing through the point P(4, 3) is y - 3 = m(x - 4), i.e. y = m(x - 4) + 3.
Plug in to the circle equation, and we get 0 = (1 + m²)x² + (6m - 2 - 8m²)x + (16m² - 24m + 5).
According to discriminant, b² = 4ac.
(6m - 2 - 8m²)² = 4(1 + m²)(16m² - 24m + 5), i.e. m = (9 ± √65)/4.
Plug in y = (9 ± √65)(x - 4)/4 + 3 to the circle equation to get A((11 minusorplus √65)/6, (5 ± √65)/6).
The length of AP is √13.

3. Hi johnny